Relating Physical Observables in QCD without Scale-Scheme Ambiguity

SLAC–PUB–5938 October 1992 (T/E)

arXiv:hep-ph/9211254v1 16 Nov 1992

Relating Physical Observables in QCD without ScaleScheme Ambiguity⋆ Hung Jung Lu Department of Physics, University of Maryland, College Park, Maryland 20742 Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford, CA 94309 PACS numbers: 11.10.Gh, 11.15.Bt, 12.38.Bx, 13.65.+i Abstract We discuss the St¨ uckelberg-Peterman extended renormalization group equations in perturbative QCD, which express the invariance of physical observables under renormalization-scale and scheme-parameter transformations. We introduce a universal coupling function that covers all possible choices of scale and scheme. Any perturbative series in QCD is shown to be equivalent to a particular point in this function. This function can be computed from a set of first-order differential equations involving the extended beta functions. We propose the use of these evolution equations instead of perturbative series for numerical evaluation of physical observables. This formalism is free of scale-scheme ambiguity and allows a reliable error analysis of higher-order corrections. It also provides a precise definition for ΛMS as the pole in the associated ’t Hooft scheme. A concrete application to R(e+ e− → hadrons) is presented. Submitted to Physical Review D ⋆ Work

supported in part by Department of Energy contract DE–AC03-76SF00515 (SLAC) and contract DE-FG05-87ER-40322 (Maryland).

The scale-scheme ambiguity problem [1,2,3] remains as one of the major cornerstones impeding precise QCD predictions. Although all physical predictions in QCD should in principle be invariant under change of renormalization scale and scheme, in practice this invariance is only approximate due to the truncation of their perturbative series. Consider the N-th order expansion series of a physical observable R in terms of a coupling constant αS (µ) given in scheme S and at a scale µ:

RN = r0 αSp (µ) + r1 (µ)αSp+1(µ) + · · · + rN (µ)αSp+N (µ) .

(1)

The infinite series R∞ is renormalization scale-scheme invariant. However, at any finite order, the scale and scheme dependencies from the coupling constant αS (µ) and from the coefficient functions ri (µ) do not exactly cancel, which leads to a remnant dependence in the finite series. Different choices of scale and scheme then lead to different theoretical predictions. The availability of next-to-next-to-leading order results in QCD [4,5] has accentuated the need for study on the scale-scheme dependence (e.g., see Ref. [6]). There have been traditionally two positions on this subject. The first one is to consider the scale-scheme ambiguity as intrinsically unavoidable, and interpret the numerical fluctuations coming from different scale and scheme choices as the error in the theoretical prediction. This point of view, aside from being overly pessimistic, is also very unsatisfactory. First of all, in general we do not know how wide a range the scale and scheme parameters should vary in order to give a correct error estimate. Secondly, besides the error due to scale-scheme uncertainties there is also the error from the omitted higher-order terms. In such an approach, it is 1

not clear whether these errors are independent or correlated. The error analysis in this context can become quite arbitrary and unreliable. A second approach is to optimize the choice of scale and scheme according to some sensible criteria. Commonly used scale setting strategies include the Principle of Minimum Sensitivity [1] (which also optimizes the choice of scheme), the Fastest Apparent Convergence criterion [2] and the BLM method [3]. In this paper we propose the use of the Extended Renormalization Group Equations as a transparent solution to the scale-scheme ambiguity problem. In this approach, a perturbative series only serves as an intermediate device for the identification of scale and scheme parameters. The ultimate prediction is obtained through evolution equations in the scale- and scheme-parameter space. This approach sets the ground for a reliable error analysis and also provides a precise definition for ΛMS . We will consider the case of QCD with Nf massless quarks. Let us first explain the concept of the universal coupling function in QCD. It is well-known that at the renormalization stage a particular subtraction prescription and a particular renormalization scale must be specified. Let us parametrize the subtraction prescription by an infinite set of continuous “scheme parameters” {ci } and the renormalization scale by µ. The universal coupling function (see Fig. 1) is the extension of an ordinary coupling constant to include the dependence on the scheme parameters {ci }: α = α(µ/Λ, {ci}) .

(2)

For the moment, the presence of the quantity Λ in the previous expression can be justified on dimensional grounds. We will identify it later with the ’t Hooft scale 2

of the chosen scheme. Stevenson [1] has shown that one can identify the beta-function coefficients of a renormalization scheme as its scheme parameters. That is, if a given scheme has the following beta-function expansion:

 α 2 α  α 3  α 4 d = −β0 − β1 − β2 +··· ; β(α) = d log µ2 4π 4π 4π 4π

(3)

then, the coefficients {βi , i = 2, 3, . . .} can be considered as the corresponding scheme parameters. (The scheme invariance of the first two coefficients β0 and β1 is a well-known fact. It is important not to confuse the coefficients βi with the β(i) functions to be introduced later.) It will be very convenient to use the first two coefficients of the beta functions to rescale the coupling constant and the scale parameter log µ2 . Let us define the rescaled coupling constant and the rescaled scale parameter as a=

β1 α , β0 4π

τ=

2β02 log(µ/Λ) . β1

(4)

Then, the rescaled beta function takes the canonical form:

β(a) =

da = −a2 (1 + a + c2 a2 + c3 a3 + · · ·) , dτ

(5)

with cn = βn β0n−1 /β1n for n = 2, 3, · · ·. This rescaling process serves to “unitarize” the expansion coefficients. For a well-behaved scheme in QCD, we would expect its beta-function expansion to roughly resemble a geometrical series, at least for = β2MS β0 /β12 , the first few coefficients. In fact, for the MS scheme we have cMS 2 3

where [7] 2 β0 = 11 − Nf , 3 38 β1 = 102 − Nf , 3 2857 5033 325 2 − Nf + N , β2MS = 2 18 54 f

(6)

and for Nf = 0, 1, 2, 3, 4, 5, 6 we have respectively c2MS = 1.5103, 1.4954, 1.4692, is of order 1.4147, 1.2851, 0.92766, −0.33654. We can clearly see that indeed cMS 2 of magnitude unity. This should be contrasted with the large value of β2MS before the rescaling process. The Extended Renormalization Group Equations simply express the invariance of physical quantities under scale- and scheme-parameter transformations. These equations have been studied long ago by St¨ uckelberg and Peterman [8] and later by Stevenson [1]: δR ∂R ∂R =β + =0, δτ ∂a ∂τ ∂R ∂R δR = β(n) =0. + δcn ∂a ∂cn

(7)

Various quantities in these equations need explanation. First of all, notice the distinction between δ/δτ and ∂/∂τ . The partial derivative δ/δτ takes into account the full variation of R under τ transformation, whereas the partial derivative ∂/∂τ affects only the expansion coefficients of R. An analogous distinction holds between δ/δcn and ∂/∂cn . In other words, in the left-hand side R = R(τ, {ci }) ,

(8)

R = R(a, rn (τ, {ci })) .

(9)

whereas in the right-hand side

4

Mathematically these are two different functions since their domains are different. However, it is common practice to use the same notation for both and draw the difference only for their partial derivatives. The fundamental beta function that appears in Eqs. (7) is defined as:

β(a, {ci }) ≡

δa = −a2 (1 + a + c2 a2 + c3 a3 + · · ·) , δτ

(10)

and the extended or scheme-parameter beta functions are defined as:

β(n) (a, {ci }) ≡

δa . δcn

(11)

As shown by Stevenson [1], these extended beta functions can be expressed in terms of the fundamental beta function. Indeed, the commutativity of second partial derivatives δ2a δ2a = δτ δcn δcn δτ

(12)

δβ(n) δβ = , δτ δcn

(13)

′ ββ(n) = β(n) β ′ − an+2 ,

(14)

implies

′ = ∂β /∂a and β ′ = ∂β/∂a. From here where β(n) (n)

β

−2



β(n) β

′

= −an+2 ,

β(n) (a, {ci }) = −β(a, {ci })

Za 0

5

xn+2 , dx 2 β (x, {ci })

(15)

where the lower limit of the integral has been set to satisfy the boundary condition β(n) ∼ O(an+1 ) .

(16)

That is, a change in the scheme parameter cn can only affect terms of order an+1 and higher in the evolution of the universal coupling function [1]. The extended renormalization group equations in (7) can be put into the form: ∂R ∂R = −β , ∂τ ∂a ∂R ∂R = −β(n) . ∂cn ∂a

(17)

These equations can now be interpreted in the following manner. The left-hand sides represent the variation of the expansion coefficients of R under scale-scheme transformations. Thus, a given perturbative series can be evolved into another perturbative series, provided we know the extended beta functions on the righthand side. In fact, in the expansion series of R as given in Eq. (1), the only quantities that cannot be modified under scale-scheme transformations are the tree-level coefficient r0 and exponent p, since they are renormalization scale- and scheme-independent. We can standardize a perturbative series R = r0 ap +· · · by defining an effective charge aR (see Grunberg in Ref. [2]) by: aR ≡



R r0

1/p

.

(18)

Since R, r0 and p are all renormalization scale and scheme invariant, the effective charge aR is also scale and scheme invariant. We only need to study the evolution of one effective charge to another. The appropriate values of r0 and p can always be put back at the end of the analysis. 6

More concretely, given two perturbative series: R = r0 apR = r0 ap + r1 ap+1 + · · · , ′

(19)

R′ = r ′ 0 apR′ = r ′ 0 ap + r ′ 1 ap +1 + · · · , ′

′

we can evolve R into R′ by following the next three steps 1) Obtain the effective charge of R aR =



R r0

1/p

r1 2 =a+ a + p r0



r2 1 − p r12 + p r0 2 p2 r02



a3 + · · · .

(20)

2) By using the extended renormalization group equations, we can change the perturbative coefficients of aR and evolve it into aR′

r′ 1 = a + ′ ′ a2 + p r0

2

1 − p′ r ′ 1 r′2 + p′ r ′ 0 2 p′ 2 r ′ 20

!

a3 + · · · .

(21)

3) Obtain R′ from aR′ by ′

R′ = r ′ 0 apR′ .

(22)

Hence, by using the extended renormalization group equations and by appropriately raising the power and rescaling our final result, we can evolve a given perturbative series R into any other perturbative series R′ . The commutativity of second partial derivatives guarantees that the final result is independent of the path chosen for evolution. Since the universal coupling functions a(τ, {ci }) covers all possible choices of scale and scheme, any effective charge aR can be expressed in terms of it: aR = a(τR , {cR i }) ,

(23)

where τR and cR i are respectively the scale and scheme parameters of R. Notice that this is also true for multiple-scale processes: given the perturbative series of a 7

multiple-scale process, we can also define an effective charge associated to it, and this effective charge can then be written in terms of the universal coupling function at a particular set of values of τ and {cR i }. The universal coupling function is dictated by the evolution equations: δa = β(a, {ci }) = −a2 (1 + a + c2 a2 + · · ·) , δτ Za xn+2 δa = β(n) (a, {ci }) = −β(a, {ci }) dx 2 . δcn β (x, {ci })

(24)

0

We shall define a(τ, {ci }) here with the boundary condition: a(0, {0}) = ∞.

(25)

Notice that the above equations contain no explicit reference to QCD parameters such as the numbers of colors or the number of flavors. Therefore, aside from its infinite dimensional character, a(τ, {ci }) is just a mathematical function like, say, Bessel functions or any other special function. Truncation of the fundamental beta function simply corresponds to evaluating a(τ, {ci }) in a subspace where higher order ci are zero. In principle, this function can be computed to arbitrary degree ∗

of precision, limited only by the truncation of the fundamental beta function.

Notice that the ’t Hooft scheme [9] defined by a′ tH (τ ) ≡ a(τ, {0}) is privileged since it is totally devoid of higher-order corrections. In fact, a′ tH (τ ) is exactly given by the solution of 1 a′ tH

+ log



a′ tH 1 + a′ tH



=τ .

(26) ′

For any single-scale process R(µ) there exists a scale µ = ΛRtH for which the ∗ The n! divergence expected for the beta function coefficients of physical effective charges can impose some theoretical limit to the achievable precision. Further discussion in this direction is beyond the scope of this paper. 8

′

′

scale parameter τR = 2β02 β1−1 log(µ/ΛRtH ) vanishes. We will call ΛRtH the ’t Hooft † scale of the R-scheme. To understand the meaning of the ’t Hooft scale, let us consider the MS scheme coupling constant:

aMS (µ) = a



′ 2β02 tH log(µ/ΛMS ), {cMS i } β1



.

(27)

′

tH : it could Notice that a priori we do not know the behavior of aMS (µ) at µ = ΛMS ′

tH is the pole in the be infinite, finite, or simply not well-defined. However, µ = ΛMS ⋆

’t Hooft scheme associated to the MS scheme:

a′ tH−MS (µ) ≡ a



′ 2β02 tH ), {0} log(µ/ΛMS β1



,

(28)

because a(0, {0}) = ∞ by boundary condition. Since the ’t Hooft scheme is completely free of higher-order corrections, this provides a precise definition for ΛMS . This cures the well-known arbitrariness in the definition of ΛMS due to the presence of higher order corrections in the MS scheme. Given two effective charges, say  ′ 2β02 tH MS aMS (µ) = a log(µ/ΛMS ), {ci } , β1  2  ′ 2β0 tH R aR (Q) = a log(Q/ΛR ), {ci } , β1 

(29)

we can expand aR (Q) in a power series of aMS (µ). To this end, we can perform a † For multiple-scale processes, the sub-manifold where the scale parameter vanishes defines the “’t Hooft surface”. ′ ⋆ There are infinite ’t Hooft schemes, differing only by the value of the ’t Hooft scale Λ tH . The word “associated” here means we are choosing the particular ’t Hooft scheme that ′ ′ tH shares the same ’t Hooft scale with the MS scheme: Λ tH = ΛMS . 9

Taylor expansion around the point

(τ, {ci }) =



′ 2β02 tH ), {cMS log(µ/ΛMS i } β1



,

(30)

and generate a series representation for aR (Q):  ′ 2β02 tH MS aR (Q) = a log(µ/ΛMS ) + τ¯, {ci + c¯i } β1     δa δa =a+ τ¯ + c¯n δτ δcn  2   2    2  1 δ a δ a δ a 2 + τ¯c¯n + c¯n c¯m τ¯ + 2 2! δτ 2 δτ δcn δcn δcm  3   δ a 1 3 τ¯ + · · · + · · · , + 3! δτ 3

(31)

i ′ ′ 2β02 h tH tH log(Q/ΛR ) − log(µ/ΛMS ) , τ¯ = β1

(32)



where

MS c¯n = cR n − cn ,

and a and its derivatives are evaluated at the point specified in Eq. (30). To order a4 , we only need the following partial derivatives: 

 δa = β = −a2 − a3 − c2 a4 + O(a5 ) , δτ   δa = β(2) = a3 + O(a5 ) , δc2   δa 1 = β(3) = a4 + O(a5 ) , δc3 2  2  δ a = 2a3 + 5a4 + O(a5 ) , δτ 2  2  δ a = −3a4 + O(a5 ) , δτ δc2  3  δ a = −6a4 + O(a5 ) . δτ 3 10

(33)

After grouping all the terms in powers of a = aMS (µ), we obtain:
aR (Q) = a − τ¯a2 + c¯2 − τ¯ + τ¯2 a3   5 2 1 3 c¯3 − (c2 + 3¯ c2 )¯ τ + τ¯ − τ¯ a4 + 2 2

(34)

+ O(a5 ) ,

where τ¯ and c¯n are as given in Eq. (32), and c2 = cMS 2 . Although we have used R and MS schemes in the derivation of this last formula, naturally it is also valid for any other pair of effective charges. Notice the occurrence of τ¯ and c¯i in all higher order coefficients. By using the evolution equations in (24), we are effectively performing a partial resummation of the perturbative series to all orders. Any physical quantity R calculated perturbatively in the MS scheme can be put into the above standard form. Hence, we can use Eq. (34) to identify the scale and scheme parameters of the effective charge of R. In other words, to give the numerical prediction for R = r0 ap + · · · we proceed to 1) compute its perturbative series in terms of the coupling constant in some scheme, say MS scheme, 2) standardize the series and identify the scale and scheme parameters (τR , {cR i }) order-by-order via Eq. (34), ′

tH , evolve the universal coupling function a(τ, {c }) 3) from the knowledge of ΛMS i

to the point (τR , {cR i }). Put back the values of r0 and p if necessary. ′

tH from the Naturally we can also go in the opposite direction and obtain ΛMS

experimental measurement of R. In Fig. 2 we show the various experimental and ′

tH , the input theoretical errors involved in the analysis. For the measurement of ΛMS

experimental error must be combined with the scheme uncertainty to give the error 11

′

′

tH . Similarly, for the prediction of a (τ ) the error from Λ tH must estimate for ΛMS R MS

be combined with the scheme uncertainty in order to give the prediction error. As a concrete example, let us consider the total hadronic cross section in e+ e− annihilation R(Q) = R(e+ e− → hadrons) recently calculated to order α3 [4,5]. For five light-quark flavors we have   α 3   α 2 α 11 1 + + 1.4092 − 12.8046 R(Q) = 3 π π π   11 αR (Q) ≡ 1+ , 3 π

(35)

where we have used α = αMS (Q) for the strong coupling constant. By putting R(Q) into the standard form of Eq. (34): aR (Q) = a + 1.1176 a2 − 8.05426 a3 ,

(36)

where a = aMS (Q), we can identify τ¯ = −1.1176 , ′

c¯2 = −10.4209 .

(37)

′

tH /Λ tH ) and c MS Knowing that τ¯ = 2β02 β1−1 log(ΛMS ¯2 = cR 2 − c2 , we conclude that R ′

′

tH ΛRtH = 1.4443 ΛMS ,

cR 2 = −9.4932 .

(38)

Experimentally [10] we have r(31.6 GeV) =

3 R(31.6 GeV) = 1.0527 ± 0.0050 , 11

(39)

which gives aR (31.6 GeV) = 0.0665 ± 0.0063 .

(40)

We have to take into account the scheme uncertainty in addition to the ex′

tH (see Fig. 3). perimental error in order to quote a correct error estimate for ΛMS

12

The scheme uncertainty of R can be quantified by a reasonable estimate of its next scheme parameter: cR 3 . G. B. West [11] has put an estimate r4 = −158.6 for the coefficient r4 in R=3

X

Q2f

   α 2  α 3  α 4 α + r2 + r3 + r4 +··· . 1 + r1 π π π π

(41)

By reducing this equation into the standard form of Eq. (34), we arrive to a value MS = −99.474. Assuming cMS is of order unity, we conclude that c¯3 = cR 3 − c3 3 R c ∼ 100. Although we have some reservation on West’s estimate (see Ref. [12]), 3

we shall nonetheless use it to illustrate our procedure. A better estimate of cR 3 will ′

tH . lead to a better error estimate for ΛMS

In Fig. 3 we show the universal charge for a0 (τ ) = a(τ, {c2 = cR 2 , c3 = c4 = . . . = 0}) and its evolution under a scheme uncertainty c3 = ±100 to a± (τ ) = a(τ, {c2 = cR 2 , c3 = ±100, c4 = c5 = . . . = 0}). The evolution in the scheme parameters are dictated by: δa = β(2) = −β δc2

Za

x4 dx 2 ∼ a3 + O(a5 ) , β

δa = β(3) = −β δc3

Za

x5 1 dx 2 ∼ a4 + O(a5 ) , β 2

0

(42)

0

For our region of interest (a ∼ 0.07) the first term in each expansion series suffices. But we should use the full integro-differential equation whenever we want to evolve a to a higher-value region. This will not only improve the accuracy of our result, but also will respect the commutativity of the second-order partial derivatives of a and thus ensure the independence of the result on the choice of integration path. To obtain a0 and a± defined above, we follow the next steps: 13

1.− Generate the ’t Hooft scheme coupling a′ tH by solving iteratively a′ tH =

1 . τ + log (1 + 1/a′ tH )

(43)

2.− Evolve a′ tH to a0 by displacing in c2 . From (42) we have: a0 =

a′ tH 1−2

cR 2

′ tH 2

a


1/2 .

(44)

3.− Evolve a0 to a± by displacing in c3 . From (42) we have: a0

a± = 1∓

3 2

3 cR 3 a0


1/3 .

(45)

In Fig. 3 we show the various errors involved in this analysis. Numerically we find the experimental, scheme, and total errors for τ to be: ∆τexp = (τ6 − τ2 )/2 = 1.41 , ∆τsch = (τ5 − τ3 )/2 = 0.22 ,

(46)

∆τtot = (τ7 − τ1 )/2 = 1.63 . ′

tH since there is a one-to-one These errors can be translated into uncertainties in ΛMS ′

tH . We can see that most error comes from the correspondence between τ and ΛMS

experimental error in aR . We can also see that the experimental error and the scheme error are highly uncorrelated since ∆τtot ∼ ∆τexp + ∆τsch . Numerically we obtain τ1 = 10.129, τ4 = 11.666 and τ7 = 13.379. Knowing that ! 2β02 31.6 GeV τ= log , ′ tH β1 1.4443 ΛMS

(47)

′

tH : we arrive at the following result for ΛMS ′

tH ΛMS = 470+310 −200 MeV .

(48)

If there were no experimental error, the estimated scheme uncertainty would lead 14

⋆

tH = 470+40 MeV . to ΛMS −30 ′

As a second application of our formalism, we will show next how to use the experimental result of aR (31.6 GeV) to predict other effective charges. Specifically, we will give a prediction for aMS (MZ ), where MZ = 91.173 GeV is the mass of the Z-boson. The evolution of aR (31.6 GeV) to aMS (MZ ) is illustrated in Fig. 4. Notice that the experimental and the scheme uncertainties confine the correct result for aR into an approximate parallelogram ABCD. We can then evolve this parallelogram into any other scheme and scale. We will use cR 3 = ±100 and cMS = ±1 to estimate the scheme uncertainties in aR and aMS . For aMS (MZ ), 3 the parallelogram ABCD is evolved into the parallelogram A′ B ′ C ′ D ′ . Notice the inversion of the orientation of the new parallelogram due to the opposite signs of MS cR 2 and c2 . Notice also the absence of scheme uncertainty in the ’t Hooft scheme.

From τA = 10.129 and τC = 13.379 and knowing that τMS = τR − τ¯ with   τ¯ = 2β02 β1−1 log(MZ /ΛMS ) − log(31.6 GeV/ΛR ) = 4.339, we find τA′ = 14.468 and τC ′ = 17.718. From here and using cMS = 0.92766 and cMS = 1 in the MS 2 3 ′

′

version of the Eqs. (44) and (45), we obtain: aA = 0.05772 and aC = 0.04818. Hence, we arrive at the prediction

aMS (MZ ) = 0.0530 ± 0.0048 ,

(49)

† or equivalently, ′

tH = 472+310 ⋆ We have rounded off the above results. More precise values are ΛMS −204 MeV ′

tH = 472+35 for the case including experimental error, and ΛMS −33 MeV for the case without experimental error. † This value is higher than the world average αMS (MZ ) = 0.1134±0.0035 quoted in the Review of Particle Properties [13] but still consistent with other quoted values for αMS (MZ ). For instance, αMS (MZ ) = 0.118 ± 0.008 is obtained by OPAL [14]. The detailed analysis of consistency between the various experimental results is beyond the purpose of this paper.

15

αMS (MZ ) = 0.132 ± 0.012 .

(50)

Let us show next that our formalism is closely related to the FAC (Fastest Apparent Convergence) criterion when only the next-to-leading-order coefficient is known. We define FAC here as the condition of a vanishing next-to-leading-order coefficient. Given aR (Q) = aMS (µ) − τ¯a2MS (µ) ,

(51)

h i ′ ′ tH τ¯ = 2β02 β1−1 log(Q/ΛRtH ) − log(µ/ΛMS ) ,

(52)

with

and assuming a complete lack of knowledge of the scheme parameters {cMS i } and {cR i }, we cannot do much better than to approximate:  ′ 2β02 tH aR (Q) = a log(Q/ΛR ), {0} , β1   2 ′ 2β0 tH log(µ/ΛMS ), {0} , aMS (µ) = a β1 

(53)

and absorb the uncertainty from scheme-parameters into our theoretical error. However, the last two equations imply that: ′

aR (Q) = aMS

µ=

tH ΛMS

ΛRtH ′

!

Q

.

(54)

Hence, by setting the coefficient τ¯ in Eq. (51) to zero, we obtain the correct ratio for the two scales. Actually, this holds true in general: by applying the FAC criterion, we always obtain the correct ratio of the ’t Hooft scales, whether or not we know the scheme parameters. Thus, despite its apparent na¨ıveness, FAC 16

actually constitutes a correct first step towards the elimination of scale-scheme ambiguity. Finally let us comment on the definition of effective charges. The definition of p is not the only possibility [15]. In fact, we can apply the extended αR by R ≡ r0 αR

renormalization group technique to the effective BLM charge [3] in the following manner. Given two physical quantities R and R′ computed in a particular scheme R(Q) = r0 (Q)αp (µ) + (r10 (Q) + r11 (Q, µ)β0 ) αp+1 (µ) + · · · ,
′ ′ ′ ′ R′ (Q′ ) = r0′ (Q′ )αp (µ′ ) + r10 (Q′ ) + r11 (Q′ , µ′ )β0 αp +1 (µ′ ) + · · · ,

(55)

where β0 = 11 − 23 Nf is the first beta function coefficient, we can define their “effective BLM charges” by p+1 p (Q) + · · · , (Q) + r10 (Q)αR−BLM R(Q) = r0 (Q)αR−BLM ′

′

R (Q ) =

′ p′ ′ ′ ′ ′ p +1 r0′ (Q′ )αR ′ −BLM (Q ) + r10 (Q )αR′ −BLM (Q ) + · · ·

(56) ,

with αR−BLM (Q) = α(µ⋆ ) where µ⋆ is the solution of r11 (Q, µ⋆ ) = 0, and similarly ′ (Q′ , µ′⋆ ) = 0. With this αR′ −BLM (Q′ ) = α(µ′⋆ ) where µ′⋆ is the solution of r11

choice of scale, vacuum polarization contributions are associated with the charge rather than the expansion coefficients, and the scale tends to reflect the mass of the virtual gluons. We can then apply the evolution equations to αR−BLM (Q) and evolve it to αR′ −BLM (Q′ ). Alternative definitions of effective charges have also been discussed in Ref. [15]. However, as we have shown in this paper any convenient choice of effective charge can be used to relate physical observables. In p practice we can adhere to the definition R ≡ r0 αR which has the advantage that

r0 and p are renormalization scale-scheme invariant quantities. 17

To summarize, we have explained the use of extended renormalization group equations to relate physical observables. The most distinctive feature of this formalism is that, in this approach, the perturbative series of a physical observable only serves to identify the scale and scheme parameters. The final prediction is obtained by the evolution of a universal coupling function. The prediction is scalescheme independent in the sense that given the initial perturbative series in any scheme at any scale, we will always obtain its correct scale and scheme parameter and hence arrive at the same prediction. We have shown that this formalism sets the ground for a reliable error analysis, and that ΛMS can be unambiguously defined as the pole in the associated ’t Hooft scheme. Finally, we have shown that this formalism is equivalent to the fastest apparent convergence criterion in the absence of information on scheme parameters.

ACKNOWLEDGEMENTS

This work was supported by Department of Energy contracts DE–AC0376SF00515 and DE–FG05–87ER–40322.

18

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[13] Particle Data Group, Review of Particle Properties, Phys. Rev. D 45, S1 (1992). [14] M. Z. Akraway et al., Z. Phys. C 49, 375 (1991). [15] J. Ch´yla, Phys. Rev. D 40, 676 (1989), G.Grunberg, ibid. 40, 680 (1989).

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FIGURE CAPTIONS 1) Pictorial representation of the universal coupling function a(τ, {ci }), where τ is the scale parameter and {ci } the scheme parameters (see definitions in text). 2) Graphical representation of the various errors involved. For the measurement ′

′

tH (or equivalently Λ tH ) the input experimental error must be combined of ΛMS R ′

tH with the scheme uncertainty. For the prediction of aR (τ ), the error in ΛMS

must be combined with the scheme uncertainty. ′

tH from the experimental result of a (31.6 GeV). We have 3) Measurement of ΛMS R

parametrized the scheme uncertainty with a value cR 3 = 100. The scheme, experimental and total errors are respectively given by ∆sch = (τ5 − τ3 )/2, ∆exp = (τ6 − τ2 )/2 and ∆tot = (τ7 − τ1 )/2. There is a one-to-one relationship ′

′

tH given by τ = 2β 2 β −1 log(31.6 GeV/1.4443 Λ tH ). between τ and ΛMS 0 1 MS

4) Prediction of aMS (MZ ) from the experimental result of aR (31.6 GeV). By using the extended renormalization group equations, the quasi-parallelogram ABCD is evolved into the quasi-parallelogram A′ B ′ C ′ D ′ . Notice the inversion of the orientation of the parallelograms due to the opposite signs of MS cR 3 and c3 . Notice also the absence of scheme uncertainty in the ’t Hooft

scheme.

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